Osteoporosis induced by cellular senescence: A mathematical model

Osteoporosis is a disease characterized by loss of bone mass, where bones become fragile and more likely to fracture. Bone density begins to decrease at age 50, and a state of osteoporosis is defined by loss of more than 25%. Cellular senescence is a permanent arrest of normal cell cycle, while maintaining cell viability. The number of senescent cells increase with age. Since osteoporosis is an aging disease, it is natural to consider the question to what extend senescent cells induce bone density loss and osteoporosis. In this paper we use a mathematical model to address this question. We determine the percent of bone loss for men and women during age 50 to 100 years, and the results depend on the rate η of net formation of senescent cell, with η = 1 being the average rate. In the case η = 1, the model simulations are in agreement with empirical data. We also consider senolytic drugs, like fisetin and quercetin, that selectively eliminate senescent cells, and assess their efficacy in terms of reducing bone loss. For example, at η = 1, with estrogen hormonal therapy and early treatment with fisetin, bone density loss for women by age 75 is 23.4% (below osteoporosis), while with no treatment with fisetin it is 25.8% (osteoporosis); without even a treatment with estrogen hormonal therapy, bone loss of 25.3% occurs already at age 65.


Comment 2
A second point on page 3 is that it is unclear what the authors mean by deficiency of androgen receptors leading to estrogen deficiency.Note also that most evidence now indicates that the majority of the skeletal effects of androgens even in men are mediated via conversion to estrogen and via the estrogen receptor (for a review, see PMID: 28710257).They should perhaps factor this into their models.
Response.We thank for the Reviewer for the comment.We agree with the Reviewer and have now added a note in Page 3, Paragraph 6 (in blue): "Note, however, that estrogen deficiency may lead to reduced bone mass through a different mechanism than loss of the androgen receptor [44]."

Response to Reviewer 2
This is a very interesting paper.Authors attempted to consider positive and negative factors of bone formation and osteoporosis as well as cell senescence using mathematical modeling methods.The results were able to predict bone loss according to aging as well as benefit of the senolytic drug.However, I think the paper sections need to be reorganized according to Journal format.

Comment 1
a. Some of the introduction parts can be moved to "methods and modeling" section.b. Figure 1 should be in modeling section and serve as hypothesis of the mathematics modeling.
Response.We thank the Reviewer for the comments.We have now reformatted the article as per the Journal style.We have also included Methods and Modeling section as suggested, and included Figure 1  c.The conclusion reads like discussion.Authors should consider adding a discussion section and only put the summary of this paper as a conclusion.
Response.We have now renamed our former Conclusion "Discussion", and added a "Conclusion" that itemizes the major findings of our work.The Conclusion now reads as follows:

Conclusion
In this paper we developed a mathematical model of aging bone with emphasis on the effect of senescence on loss of bone density: Our model simulations of bone density loss for aging people are in agreement with standard charts for normal healthy men and women.We used the model to estimate the effect of a senolytic drug, a drug which eliminates senescent cells, on reducing bone density.We quantified, in Tables, the benefits of early treatment with senolytic drugs, for men and women.
In particular, women taking estrogen hormonal therapy and starting early treatment with senolytic drug can delay a state of osteoporosis by 10 years.
In this paper we focused on osteoporosis induced by senescence, and the effect of senolytic drugs in slowing the progression to osteoporosis.There are, of course, several standard drugs in the treatment of osteoporosis, such as Denosumab and BP Alendronate.It would be interesting to study therapy that combines such a drug with a senolytic drug (e.g., fisetin or quercetin) once the senolytic drugs have been successful in clinical trials.

Other comments
Following some comments that need to be revised.1. Table 6 can be moved to the methods section and show earlier so that the readers know what each variable represents in the equation.
and Table 6 (now Table 2) in the Modeling section.
Table 6 (now Table 2) in the Modeling section (Page 5).